Extruded Part disappears after switching from Edit Mode into Object Mode
Shape Keys

I just deleted the Shape Keys for Head_Cube.005 and now the extruded arm face stays throughout the animation:  



I don't know any more about Shape Keys, but here is a link to them on the Blender wiki:
http://wiki.blender.org/index.php/Special:Search?search=shape+keys&amp;fulltext=

If you need the shape keys, then you will need to figure out how to edit them to incorporate the extrusion throughout the animation. I'd help more with them if I could, but I know nothing else about them.